1. Field of the Invention
This invention concerns enhancing the resolution of optical imaging systems. This invention relates to imaging interferometers that create an image of a laterally extended object.
2. Discussion of Related Art
There are two kinds of imaging systems, those that form an image directly and those that synthesize the image from its Fourier components. When an object is illuminated with electromagnetic radiation it creates a diffraction pattern. If the radiation is spatially coherent, then the diffraction consists of a summation of plane waves diverging away from the object each representing one spatial frequency component of the electric field distribution on the object. The angle of diffraction of a particular wave is equal to the spatial frequency times the wavelength of light. Each one of these waves has amplitude and phase associated with it. On a screen in the far field, the diffraction produces an irradiance pattern, which is identical to the magnitude squared of the Fourier Transform of the electric field leaving the object. The phase information is lost. On the other hand if the body is illuminated with incoherent radiation or if it is self luminous, such as distant stars, then the illumination is uniform but the degree of coherence in the aperture plane is given by the Fourier Transform of the irradiance distribution on the object, according to the van Cittert-Zernike theorem, which is used extensively in astronomy.
Conventional imaging systems, such as a lens or holography, aim at capturing and recombining most of the waves that are diffracted from the object in order to construct the image. In principle, if all the divergent waves are redirected to intersect and overlap over some region of space while maintaining their original amplitudes, relative phases and angles then they would recreate a perfect image of the object. The failure of such systems to reproduce a perfect image is mainly due to the inability to capture the high frequencies, besides imperfections in the quality of the optical surfaces. In fact, image formation is an interferometric process. A lens is capable of forming an image of a coherent as well as an incoherent object because it has the property that all the rays and waves travel equal optical paths between the object plane and image plane. Thus, it does not matter what the coherence length is, the lens will form an image on-axis because the optical path length difference there is zero. The interference takes place over the entire image space and not just on-axis. At off-axis points in the image plane the path length difference changes gradually. Thus, the ability to interfere completely off-axis depends on the degree of coherence among the waves. The absence of the high frequency waves in finite aperture imaging systems creates artifacts in the image plane, which account for the degradation of the performance of coherent imaging systems. By contrast, the lack of coherence smoothes out the off-axis intensity variations in incoherent systems.
It is desired to broaden the aperture of imaging systems in order to enhance the imaging resolution. It is also desired to capture more photons from dim or rapidly varying sources to improve the signal to noise ratio. An increase in the size of the aperture of a lens or primary parabolic mirror is impractical beyond a certain limit because the cost of fabrication of large focusing elements to optical tolerances becomes prohibitive. Holographic imaging techniques replace the lens with a holographically fabricated grating. The phase of the first diffracted order from the holographic grating varies as the quadratic of the distance off-axis. This is analogous to the phase incurred by a ray traversing a lens, which varies quadratically with the radial position of the ray. A holographically fabricated grating preserves the phase and angle relationships among the diffracted waves similar to a lens, which allows it to reproduce the image with fidelity during reconstruction. A holographic grating accomplishes by diffraction what a lens does by refraction.
Conventional imaging systems utilize a focusing element, such as lens or parabolic mirror at full aperture to image distant objects in the focal plane. The focused image is susceptible to atmospheric aberrations and to imperfections of the optical surface. In order to reduce the effect of aberrations, it is desired to defocus or spread the light in the image plane. This can be accomplished by restricting the aperture, i.e. the use of synthetic aperture techniques and non-focusing optical elements, such as planar mirrors.
Coherent Imaging
The imaging of a coherent object can be achieved using a conventional full-aperture system, such as a lens or hologram. The electric fields of the diffracted waves add in amplitude and phase when recombined in the image plane. This produces a replica of the object if the amplitude, phase and angle relationships among the waves are preserved. It is not necessary in conventional imaging systems to measure or know the magnitude or phase of the electric fields. As long as the amplitude, phase and angle relationships, corresponding to the magnification of the imaging system, are not perturbed then the optical system will reconstruct the object with high fidelity. The burden of conventional systems is capturing the high frequency components of the diffracted waves to achieve a more complete interference. This requires bigger apertures, which increase the cost of the system significantly. Coherent imaging systems used in photolithography for the fabrication of electronic circuits aim to achieve sub-wavelength resolution.
Synthetic Aperture
Unconventional systems attempt to create the effect of a large aperture synthetically by sampling a subset of the diffracted waves with the use of two or more sub-apertures. Since imaging along two axes is usually desired and to limit the displacement of each sub-aperture, several sparsely-located sub-apertures are deployed in the pupil plane. The basic technique entails the use of two very narrow apertures, e.g. pinholes as in Young's experiment, so that the diffraction effects become dominant. If the pinholes are placed in the far field of a coherently illuminated object, then each pin hole intercepts only one diffracted wave from the object. If the object is placed on the optical axis then the two pinholes capture the conjugate positive and negative frequencies of its Fourier Transform, i.e. the waves traveling symmetrically off-axis. The goal is to measure the amplitude and phase of the spatial frequency components of the field by mapping its Fourier Transform in the frequency domain. The diffraction from the pinholes creates sinusoidal fringes in the image plane. The spatial frequency of the fringes is determined by the spacing between the pinholes. The visibility of the fringes depends on the ratio of the amplitudes of the two interfering waves and the phase of the fringes represents the difference between the phases of the Fourier components. Unequal amplitudes cause a decrease in visibility or modulation of the fringes. A phase difference between the waves causes a shift of the central fringe off-axis. By varying the spacing between the holes the entire spatial spectrum of the object can be measured. It is worth noting that for a coherently illuminated object the Fourier Transform is not necessarily an even function, i.e. the magnitudes and phases of the conjugate positive and negative frequencies can differ. The object is subsequently reconstructed in the spatial domain by inverse Fourier transforming the data. Thus, synthetic aperture techniques inevitably involve computation in a two step process. Similarly, holography performs image reconstruction is a two step process, namely the write and read cycles. For this reason holography is not considered to be a real-time process. Nevertheless, the processing times can be shortened. This adds delay in the processing of the image, which becomes a concern especially for moving targets.
Incoherent Imaging
In the case of an incoherently illuminated or a self luminous object the magnitude and phase of the electric field cannot be uniquely defined. The irradiance in the aperture plane is uniform and the coherence function is given by the Fourier Transform of the irradiance of the object according to the van Cittert-Zernike theorem. Thus, the degradation in the visibility of the fringes is due to the lack of perfect coherence due to the diffraction from the pinholes of all the incident waves. The complex visibility of the central fringe, which is measured experimentally, is equal to the coherence function. The envelope of the fringes decays due to the finite coherence length of the light source and due to diffraction from the finite-width apertures. Thus, by varying the spacing between the apertures the entire spatial spectrum of the irradiance of the object can be mapped in the frequency domain, which is then inverted to obtain the irradiance distribution in the spatial domain, i.e. the image. Synthetic aperture techniques are known as Fourier imaging because the Fourier Transform of the image, rather than the image itself is obtained, which requires further computation to derive the image. By contrast, the full aperture lens and hologram are direct imaging techniques because the overlapping waves construct the image directly.
In either coherent or incoherent imaging it would be necessary to increase the size of the aperture in order to improve the imaging resolution. Since a Fourier Transform relationship exists between the object plane and the pupil plane, the resolution in one plane is inversely proportional to the total sampling interval in the other. For this reason, the apertures of the very large baseline telescope (VLBT) are pushed as far apart as possible in order to achieve nano-radian resolution. For example, it is desired to utilize apertures with diameters on the order of 10 to 30 meters to image space and astronomical objects to achieve an angular resolution of 10 to 50 nano-radians in the visible. It would be impractical and prohibitively expensive to construct a curved mirror or lens of this diameter out of a monolithic piece of glass while maintaining a high quality optical surface. The advantage of the synthetic aperture technique is that it achieves the resolution of a very large aperture with two smaller apertures positioned diametrically opposite each other.
Lensless Imaging
Imaging systems can be classified either as direct or Fourier. The lens is the only optical device that can produce a direct two-dimensional image instantaneously. Holography produces a direct image but it is a two-step process; so is synthetic aperture. However, synthetic aperture can extend the aperture beyond the limits of a lensed system. For this reason, it has been the goal of imaging system designers to eliminate the lens, especially in the push toward bigger apertures. It is worth noting, however, that even though holography replaces the lens with a grating, the synthetic aperture technique does not preclude using a lens. A lens can be masked entirely except for two pinholes, for example, and the interference pattern is transformed from a focused Airy pattern to a sinusoidal interferogram. In fact, A. A. Michelson's early experiment in 1920 atop Mount Wilson, which gave birth to stellar interferometry, consisted of a lensed synthetic aperture system. He covered most of the 10′ telescope except for two 6″ diameter holes. However, Michelson did not produce a complete imaging system. His goal was to measure the stellar diameter. He managed to observe the fringes and make quantitative measurements in spite of atmospheric turbulence, which caused the fringes to wander and drift. This demonstrated the tolerance of the synthetic aperture technique to atmospheric disturbances by virtue of spreading the light in the image plane, i.e. observing an interferogram instead of a focused image, besides the ability to position the outer mirrors at distances greater than the diameter of the telescope. Nevertheless, the potential of the synthetic aperture technique is to deliver high resolution imaging without using a lens.
Therefore, the goal is to produce a two-dimensional image and demonstrate high resolution without using a lens. Another goal is to produce as close to a direct image as possible, i.e. to display the image in real space and time by simplifying the algorithm and minimizing the computations.
Magnification
A fundamental aspect of imaging is determining the correct magnification. The overlap of waves in the image plane of a lens forms a perfect image, but the observed image cannot be related to the real object unless the distance between the object and the lens is known. In principle, the plane of the image and the magnification can be determined experimentally at the location of best focus. However, for distant objects the image plane coincides with the focal plane, and the linear magnification vanishes. If no focusing element is used then the fringes become non-localized and form in any region of overlap in the far field. The lens reproduces the far field or Fraunhofer conditions in its focal plane and provides a length scale, i.e. focal length by which off-axis distances are measured. Holography, on the other hand, does not focus or localize the fringes and usually produces an image with unit magnification if the same wavelength that was used to write the hologram is used again to read it. Holography circumvents the issue of magnification; however the real image is formed symmetrically about the plane of the hologram at equal distance from where the object originally stood. Thus, it would be impractical to image distant or space objects holographically because the image would be located far away in space in the opposite direction from the object. The synthetic aperture technique does not resolve the issue of magnification either without prior knowledge of the size of the object or its distance. Under coherent illumination the pin holes sample two conjugate frequencies from the spectrum of the electric field. The measurement of the complex visibility yields the ratio of the two amplitudes of the waves and their phase difference. But it is not possible to relate the measurement to a specific spatial frequency component or angle of diffraction unless the distance to the object is known. In the absence of such knowledge, an image resembling the object can be synthesized by inverse Fourier transformation but it would be related to the real object through an unknown scaling factor. The spatial frequency of the fringes in the image plane is not related to the spatial frequencies of the object. It depends on the spacing between the holes and the distance between the pupil and image planes. In the coherent case it is not possible to identify the central fringe because all the fringes have equal visibility, and therefore it is not possible to determine the exact magnification. In the case of incoherent illumination the measurement of visibility yields an estimate of the complex degree of coherence. It is possible to pin point the central fringe in an incoherent interferogram because of the decaying envelope of the fringes due to finite temporal coherence. However, this does not yield a direct measurement of the spatial components of the irradiance function of the object. Either the size of the object or its distance from the pupil plane must be known in order to properly characterize the object. However, for very distant objects the visibility measurement can be related to the angle subtended by the diameter of the object as observed from a point in the pupil plane. This fact is widely used in astronomy to estimate angular diameters of stars, rather than image objects at finite distances with known magnification.
Determining the scaling factor between the reconstructed image and the actual object requires knowledge of either distance to the object or to the image plane. However, in an afocal system, which does not use a lens there is no unique image plane. The fringes are non-localized and can be observed at any plane within the overlap region in the far field.
The synthetic aperture technique does not measure the phase difference between two non-conjugate orders. The inversion of the Fourier Transform necessary to reconstruct the image requires knowledge of the phase relationships among all the constituent orders. Thus, the synthetic aperture technique does not measure the amplitude and phase as a function of the angle of diffraction and second, measurement of the complex visibility does not yield enough phase relationships among the diffracted orders to permit computation of the inverse Fourier Transform.
Longitudinal and Lateral Imaging
It is desired to enhance the resolution of optical imaging systems. An object has three-dimensions, one longitudinal along the optical axis, and two lateral dimensions. It is desired to enhance the longitudinal as well as lateral resolutions. Light from an idealized point source passes through an optical system and is projected on a screen in the far field perpendicular to the optical axis. The light interferes on the screen and forms the image of the point source, which is the point spread function (PSF). Information about the point source is obtained by analyzing the PSF. Wave front dividing systems consist of one or more apertures. The PSF is the Fourier Transform of the aperture plane. Qualitatively described, the width of the light distribution in the far field is inversely proportional to the width of the apertures. The PSF of a single circular aperture is the Airy pattern, while that of a rectangular aperture is the sin(x)/x function. A typical PSF of a single aperture system has a width equal to ƒλ/a, where a and ƒ are the width and focal length of the imaging system, respectively, and λ is the wavelength of light. The PSF of a system of two apertures separated by a distance D consists of sinusoidal fringes of period ƒλ/D. The fringes are modulated by an envelope of width ƒλ/a, corresponding to the width a of each aperture. Thus, the PSF of a two-aperture system consists of D/a fringes. This is typical of diffraction-limited systems. Another factor, which affects the interference, is the degree of coherence of the light source. A perfectly temporally coherent point source produces fringes, which are only diffraction limited. A partially coherent point source yields a number of fringes, which is equal to the coherence length of the source, Lc, divided by the wavelength λ. Thus, the number of fringes is given by D/a, or Lc/λ, whichever is smaller; that is the system is either diffraction limited or coherence limited.
Wavefront Division and Amplitude Division
Wave front division (WD) imaging systems, such as the Michelson stellar interferometer, which is based on Young's Experiment, focus the light on two narrow apertures, such as slits or holes. Several apertures or sub-apertures can be used, which create a speckle pattern in the focal plane. These systems are diffraction limited. Amplitude dividing (AD) systems, such as the Mach-Zehnder or Michelson interferometer are often used to create sinusoidal fringes by interfering two collimated beams at full aperture. These systems are usually coherence limited rather than diffraction limited. It is often desired to image faint sources. For this reason the beams are usually partially focused or compressed to a narrower cross-section using optical reduction. This enhances the signal to noise ratio and improves the quality of the image. The intersecting beams subtend a half angle θ. For small angles, the period of the fringes is equal to λ/2θ or λ/(2 NA), where NA is the numerical aperture of the imaging system. The position of the fringes is determined by the phase difference between the beams. A shift in phase causes the fringes to move in the observation plane. The key to enhancing the resolution of imaging systems is tracking the motion of the fringes. Longitudinal and lateral displacements of the point source cause the fringes to move by different amounts. For example, WD systems are less sensitive to longitudinal motion because the phase difference at the two apertures remains almost unchanged. Similarly, AD systems are less sensitive to lateral motion of the mirror. For this reason, WD systems, such as the stellar interferometer and the microscope are used for lateral imaging, while AD systems, such as the Michelson and Mach Zehnder and Twyman-Green interferometers are used for longitudinal imaging with the use of a reference arm. Longitudinal imaging systems usually image a single source, whereas lateral imaging systems image multi sources.
Image of a Point Source through Two Apertures
The image of a point source through a system of two apertures is a sinusoidal fringe pattern, which has a limited lateral range. The number of fringes is determined by the diffraction from the two apertures and the temporal coherence properties of the light source. The period of the fringes is determined by the numerical aperture of the optical system and the phase is determined by the angular position of the point source relative to the optical axis. The amplitude of the fringes is determined by the brightness of the source. The central fringe corresponds to the point on the observation plane where the optical path length difference between the two arms of the interferometer is zero. The images of two point sources overlap. The phase difference between the two central fringes of two point sources is the image of the object contained between those two points. The images of several points yield overlapping sinusoidal fringes with different amplitudes and phases. The light distribution consisting of the superposition of all these fringes is recorded with a CCD array or a photodetector. Imaging of the points is equivalent to unraveling the individual phases and amplitudes.
Visibility
The light source is often faint and very distant from the optical system. The light must travel through some atmosphere before it reaches the detector. The atmosphere often interferes with and degrades the quality of the image. Specifically, the atmosphere introduces a random and time varying phase shift to the fringes. However, the relative phase shifts between the fringes and the period of the fringes are unaffected. An effective technique to cancel the effect of the drift and recover the original phases is to measure the visibility of the fringes, i.e. the ratio of intensity variation to the average optical power. The visibility of the superposition of fringes is a function of the relative phases but also depends on the amplitudes. For this reason, visibility techniques are usually limited to bright sources of equal brightness.
Synthetic Imaging Technique
Two-aperture systems have the potential for higher angular resolution than single aperture systems of comparable size because of the ability to discern a phase change of a fraction of 2π. A typical interferometric configuration that is commonly used for lateral imaging is the Michelson stellar interferometer (MSI), which is based on Young's Experiment. A schematic representation of this configuration is shown in FIG. 1. The basic concept consists of only two narrow slits, however, mirrors are added to gain sensitivity because the mirrors can be placed much farther apart. Collimated light from distant sources is focused by the system of mirrors M1 and M2 on slits S1 and S2 by mirrors M3 and M4. The optical path lengths traveled by all the rays in one arm of the interferometer are equal. For a point source located at an angle θ relative to the axis, the optical path length difference between the two arms is Lθ, where L is the separation between the outer mirrors M1 and M2. The purpose of the inner mirrors is to focus the light on the slits. The distance between the slits is chosen to yield the desired numerical aperture of the imaging system. The angle between the interfering rays at any point in the image plane is determined by the numerical aperture NA and is independent of the angle of incidence. The period of the fringes, λ/(2 NA), can be chosen arbitrarily as long as the fringes can be viewed with available photodetectors. The size of the image, i.e. the phase difference between the images of two points is proportional to the separation L. As L is increased the size of the image increases and the visibility V of the central fringe changes. V is plotted vs L. The image is obtained by taking the inverse Fourier Transform of V(L) according to the Van Cittert-Zernike theorem. Thus, the imaging resolution is inversely proportional to the sampling interval L. For this reason the mirrors of the Very Long Baseline stellar Interferometer are placed very far apart, up to hundreds of meters, in order to gain angular resolution. This is equivalent to unraveling the phases of the sinusoidal fringes in the image plane. If it is desired to image N point sources each having a different brightness, then we have 2N unknowns corresponding to the N phases and N amplitudes. Scanning the mirrors, i.e. changing L is equivalent to providing a set of 2N equations for the 2N unknowns. Thus, Fourier transforming V(L) by the Van Cittert-Zernike theorem is equivalent to solving a system of 2N equations by 2N unknowns. Obtaining the relative phases and amplitudes of the fringes corresponding to the point sources that make up an object is tantamount to imaging the object. This is known as synthetic imaging.
Sheared Wavefronts
An interferometer splits an original wavefront and then recombines it with itself. Typically the wavefront is sheared, that is a portion of the wavefront interferes with another portion of the same wavefront upon recombination. Different interferometric configurations exhibit different amounts of shear. For example, wavefront dividing interferometers exhibit no shear because the original wavefront is destroyed upon focusing. A new wavefront emerges from each slit and the shear is lost. All the rays being focused on one slit in one arm of the interferometer have equal optical path lengths. The interference beyond the plane of the apertures is governed by the diffraction from the slits. By contrast, amplitude splitting interferometers exhibit shear.
Erect and Inverted Shear
The shearing properties of interferometers are best illustrated using collimated light. Imagine that light from a point source is collimated and directed at an amplitude splitting interferometer, such as Michelson or Mach Zehnder. The incident wavefront is split by a partially reflecting/transmitting mirror along two different paths. The split beams are redirected by mirrors tilted at the proper angle to create either spatial or temporal fringes. The fringes are created at full aperture and the interference is more coherence than diffraction limited. If an imaginary line is drawn along the middle of the incident wavefront bisecting it in half, and the two split wavefronts are made to overlap completely on the detector, then depending on the number of mirrors encountered in each path each half of the original wavefront will either interfere with itself or with the other half upon recombination. If the difference between the number of reflections along the two paths is even, such as the Michelson configuration where each beam experiences two reflections, then each half will interfere with itself and the shear is erect. If on the other hand the difference is odd, such as the Mach Zehnder configuration (leaving out the exit beam combiner) where one beam encounters two mirrors along its path and the other beam only one, then each half of the original wavefront will interfere with the other half and the shear is inverted. This terminology is analogous to the imaging properties of a lens, which depending on the location of the object will either create an erect or inverted image. It is usually desirable to achieve complete overlap of the two wavefronts. However, the amount of shear can be varied by displacing the mirrors to vary the overlap between the two intersecting beams. In the case of erect shear the shear is constant across the wavefront, i.e. the separation between any two interfering rays in the original wavefront is constant. This is the case of shearing interferometers, which are commonly used to measure tilt of wavefronts. In an inverted shear configuration the shear varies across the wavefront. An arbitrary number of mirrors can be added to each path of the interferometer, which will not change its shearing properties as long as the difference remains either even or odd.
Diffractionless Interferometer
If the slits in the MSI design of FIG. 1 were removed in order to eliminate their diffraction effects and the two wavefronts were allowed to propagate and interfere at full aperture on the image plane, then we would obtain the erect shear configuration of FIG. 2. The beams could also be partially focused or reduced optically in order to limit the area of interference and raise the signal to noise ratio. Mirrors M3 and M4 are oriented slightly differently from those in FIG. 1 in order to cause the two beams to interfere at a shallow angle, for example about 0.75 degree to yield detectable fringes with a period of about 20 microns for visible light. The distance between mirrors M3 and M4 is chosen in conjunction with the distance to the image plane to yield the desired NA. The period of the fringes is not critical because the visibility is measured, which is independent of period. A change of θ in the angle of incidence corresponding to two point sources yields an optical path length difference of Lθ, which causes a phase shift of (Lθ/λ) 2π regardless of the fringe period. Thus, the phase and image vary linearly with the size of the object. Even though the angle of each beam falling on the observation plane changes with the angle of incidence, the subtended angle between the two beams remains constant because the beams track each other. This insures that the period of the fringes remains constant. However, the fringes turn and change their phase in the observation plane. The result is that two distant point sources yield identical fringe patterns that are displaced from each other by a phase shift corresponding to the image. This is similar to the stellar interferometer of FIG. 1. The imaging resolution is obtained from the analysis of the visibility V of the central fringe plotted against the outer mirror separation L. For higher resolution, L is increased. In conclusion, if diffraction plays a significant role in the interference or if the shear is erect, then the imaging is obtained from analysis of the visibility and the angular resolution is enhanced by increasing the distance between the outer mirrors.